Quantum degeneracy problem, electron on a ring

In summary, the conversation discusses the periodic infinite square well and its relationship to the Hamiltonian and Schrodinger's equation. It also touches on the degeneracy of the energy levels and the stability of compounds with different numbers of free electrons. There is still some confusion about the interpretation of stability in relation to the number of electrons.
  • #1

Homework Statement



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Homework Equations



Below

The Attempt at a Solution



So this is a lot like the infinite square well, except periodic. If S is an arc length, then [itex]S = \theta R[/itex] so [itex]\frac{d^2}{dS^2} = \frac{1}{R^2}\frac{d^2}{d\theta^2}[/itex], which is more convenient to use in the hamiltonian. So for the hamiltonian I get:

[tex]H = \frac{-\hbar^2}{2m}\frac{1}{R^2}\frac{d^2}{d\theta^2} + V_0[/tex]

With Schrodinger's equation, I get

[tex]\frac{d^2 \psi^2}{d\theta^2} = -\frac{2mR^2(E - V_0)}{\hbar^2}\psi[/tex]

Which gives solutions of the form [itex]\psi = \frac{1}{\sqrt{2\pi}}e^{\pm i k \theta}[/itex], where [itex]k = \sqrt{\frac{2mR^2(E - V_0)}{\hbar^2}}[/itex].

Then, because it's a ring, we need [itex]\psi(x + 2\pi) = \psi(x)[/itex] for any x, which gives us the requirement that k is an integer. So our energy levels are [itex]E = \hbar^2 k^2/2mR^2 + V_0[/itex], and it seems like they have a degeneracy of 4 because we have two functions for each k with the same energy, and then for each of them, the electron's spin can be up or down. Is that right?

As for part (b), I have no idea... Benzene has 6 free electrons, so according to my degeneracy, it completely fills the first energy level, and then there are 2 electrons in the 2nd energy level. Ok...then they ask about a compound with 4 electrons. This seems like it just fills the first energy level, but that seems too simple and stupid to be right.

Can anyone help me out?

Thanks!
 
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  • #2
k is an integer including zero. What is the degeneracy of the energy level corresponding to k = 0?
 
  • #3
TSny said:
k is an integer including zero. What is the degeneracy of the energy level corresponding to k = 0?

Well I guess that just has a degeneracy of 2 due to the spin, right?

Any idea on the aromatic thing? I'm totally clueless about that...
 
  • #4
Ah, I think I see, after what you just said and reading the wiki article on Huckel's Rule... The number of states for k =/= 0 is 4 for each energy level. For k = 0 it's 2. So like Huckel's Rule, for energy level n, there are 4n + 2 states. So benzene is aromatic because it has 6 electrons, so fully completes the n = 1 energy level. The other compound has 4 however, so it fills the n = 0 level but only half fills the n = 1 level, so it's not stable (though from the little I know of chemistry, I thought that just meant it's more reactive, not less stable).
 
  • #5
I think your analysis is now correct. I'm not very clear on the reactive vs. stable interpretation either. Maybe someone can clarify it.
 

1. What is the quantum degeneracy problem?

The quantum degeneracy problem refers to the phenomenon where particles, such as electrons, can exist in the same state or energy level. This is a fundamental principle in quantum mechanics and plays a crucial role in understanding the behavior of particles in systems such as atoms and molecules.

2. How does the electron on a ring model relate to the quantum degeneracy problem?

The electron on a ring model is a simplified version of the quantum degeneracy problem, where a single electron is confined to a circular path. This model allows scientists to study the effects of quantum degeneracy in a simpler and more controlled system.

3. What is the significance of studying the electron on a ring model?

Studying the electron on a ring model can provide insights into the behavior of electrons in more complex systems, such as atoms and molecules. It also allows scientists to explore the effects of quantum degeneracy and understand its role in various physical phenomena.

4. How is the electron on a ring model experimentally realized?

The electron on a ring model can be experimentally realized using semiconductor devices, such as quantum rings or quantum dots. These devices can confine electrons to a circular path, allowing scientists to observe and study their behavior.

5. What are the potential applications of understanding the quantum degeneracy problem?

Understanding the quantum degeneracy problem has numerous potential applications in fields such as quantum computing, material science, and nanotechnology. It can also lead to advancements in our understanding of fundamental physics principles and the development of new technologies.

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